Effective Dynamics of Extended Fermi Gases in the High-Density Regime

TL;DR

This paper analyzes the quantum dynamics of large Fermi gases at high density, demonstrating convergence to Hartree equations in both non-relativistic and relativistic cases, with results applicable to extensive systems.

Contribution

It establishes density-dependent convergence of many-body Fermi gas evolution to Hartree equations, extending understanding to large-scale and high-density regimes.

Findings

Convergence to Hartree equations in non-relativistic case for short times.

Convergence to relativistic Hartree equation for all macroscopic times.

Rate of convergence depends only on density, not total particle number.

Abstract

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases.